{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $\\lambda$ variable for oblate ellipsoids"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we follow the reasoning presented by [Webster (1904)](#webster-dynamics) for analyzing the ellipsoidal coordinate  $\\lambda$ describing a oblate ellipsoid. Let's consider an ellipsoid with semi-axes $a$, $b$, $c$ oriented along the $x$-, $y$-, and $z$-axis, respectively, where $0 < a < b = c$. This ellipsoid is defined by the following equation:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq1'></a>\n",
    "$$\n",
    "\\frac{x^{2}}{a^{2}} + \\frac{y^{2} + z^{2}}{b^{2}} = 1 \\: . \\tag{1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A quadric surface which is confocal with the ellipsoid defined in [equation 1](#eq1) can be described as follows:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq2'></a>\n",
    "\n",
    "$$\n",
    "\\frac{x^{2}}{a^{2} + \\rho} + \\frac{y^{2} + z^{2}}{b^{2} + \\rho}= 1 \\: , \\tag{2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\rho$ is a real number. We know that [equation 2](#eq2) represents an ellipsoid for $\\rho$ satisfying the condition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq3'></a>\n",
    "\n",
    "$$\n",
    "\\rho + a^{2} > 0 \\: . \\tag{3}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given $a$, $b$, and a $\\rho$ satisfying [equation 3](#eq3), we may use [equation 2](#eq2) for determining a set of points $(x, y, z)$ lying on the surface of an ellipsoid confocal with that one defined in [equation 1](#eq1). Now, consider the problem of determining the ellipsoid which is confocal with that one defined in [equation 1](#eq1) and pass through a particular point $(x, y, z)$. This problem consists in determining the real number $\\rho$ that, given $a$, $b$, $x$, $y$, and $z$, satisfies the [equation 2](#eq2)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "By rearranging [equation 2](#eq2), we obtain the following quadratic equation for $\\rho$:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "f(\\rho) = (a^{2} + \\rho)(b^{2} + \\rho) - (b^{2} + \\rho) \\, x^{2}\n",
    "- (a^{2} + \\rho) \\, (y^{2} + z^{2}) \\: .\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This equation shows that:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\rho = \\begin{cases}\n",
    "d \\to \\infty \\: &, \\quad f(\\rho) > 0 \\\\\n",
    "-a^{2} \\: &, \\quad f(\\rho) < 0 \\\\\n",
    "-b^{2} \\: &, \\quad f(\\rho) > 0\n",
    "\\end{cases} \\: .\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By rearranging this equation, we obtain a simpler one given by:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq4'></a>\n",
    "\n",
    "$$\n",
    "f(\\rho) = p_{2} \\, \\rho^{2} + p_{1} \\, \\rho + p_{0} \\: , \\tag{4}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq5'></a>\n",
    "\n",
    "$$\n",
    "p_{2} = 1 \\: , \\tag{5}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq6'></a>\n",
    "\n",
    "$$\n",
    "p_{1} =  a^{2} + b^{2} - x^{2} - y^{2} - z^{2} \\tag{6}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq7'></a>\n",
    "\n",
    "$$\n",
    "p_{0} =  a^{2} \\, b^{2} - b^{2} \\, x^{2} - a^{2} \\, y^{2} - a^{2} \\, z^{2} \\: . \\tag{7}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that a particular $\\rho$ satisfying [equation 2](#eq2) results in $f(\\rho) = 0$ ([equation 4](#eq4)).\n",
    "\n",
    "In order to illustrate the parameter $\\rho$, consider the constants $a$, $b$, $x$, $y$, and $z$ given in the cell below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a = 11.\n",
    "b = 20.\n",
    "x = 21.\n",
    "y = 23.\n",
    "z = 30."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By using these constants, we calculate the coefficients $p_{2}$ ([equation 5](#eq5)), $p_{1}$ ([equation 6](#eq6)) and $p_{0}$ ([equation 7](#eq7)) as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "p2 = 1.\n",
    "p1 = a**2 + b**2 - (x**2) - (y**2) - (z**2)\n",
    "p0 = (a*b)**2 - (b*x)**2 - (a*y)**2 - (a*z)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the sequence, we define a set of values for the variable $\\rho$ in an interval $\\left[ \\rho_{min} \\, , \\rho_{max} \\right]$ and evaluate the quadratic equation $f(\\rho)$ ([equation 4](#eq4))."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "rho_min = -b**2 - 500.\n",
    "rho_max = -a**2 + 2500.\n",
    "rho = np.linspace(rho_min, rho_max, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f = p2*(rho**2) + p1*rho + p0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, the cell below shows the quadratic equation $f(\\rho)$ ([equation 4](#eq4)) evaluated in the range $\\left[ \\rho_{min} \\, , \\rho_{max} \\right]$ defined above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA9wAAAGHCAYAAABChloAAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa62f128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ymin = np.min(f) - 0.1*(np.max(f) - np.min(f))\n",
    "ymax = np.max(f) + 0.1*(np.max(f) - np.min(f))\n",
    "\n",
    "plt.close('all')\n",
    "plt.figure(figsize=(10,4))\n",
    "\n",
    "plt.subplot(1,2,1)\n",
    "plt.plot([rho_min, rho_max], [0., 0.], 'k-')\n",
    "plt.plot([-a**2, -a**2], [ymin, ymax], 'r--', label = '$-a^{2}$')\n",
    "plt.plot([-b**2, -b**2], [ymin, ymax], 'g--', label = '$-b^{2}$')\n",
    "plt.plot(rho, f, 'k-', linewidth=2.)\n",
    "plt.xlim(rho_min, rho_max)\n",
    "plt.ylim(ymin, ymax)\n",
    "plt.legend(loc = 'best')\n",
    "plt.xlabel('$\\\\rho$', fontsize = 20)\n",
    "plt.ylabel('$f(\\\\rho)$', fontsize = 20)\n",
    "\n",
    "plt.subplot(1,2,2)\n",
    "plt.plot([rho_min, rho_max], [0., 0.], 'k-')\n",
    "plt.plot([-a**2, -a**2], [ymin, ymax], 'r--', label = '$-a^{2}$')\n",
    "plt.plot([-b**2, -b**2], [ymin, ymax], 'g--', label = '$-b^{2}$')\n",
    "plt.plot(rho, f, 'k-', linewidth=2.)\n",
    "plt.xlim(-600., 100.)\n",
    "plt.ylim(-0.3*10**6, 10**6)\n",
    "plt.legend(loc = 'best')\n",
    "plt.xlabel('$\\\\rho$', fontsize = 20)\n",
    "#plt.ylabel('$f(\\\\rho)$', fontsize = 20)\n",
    "\n",
    "plt.tight_layout()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remember that we are interested in a $\\rho$ satisfying [equation 3](#eq3). Consequently, according to the figures shown above, we are interested in the largest root $\\lambda$ of the quadratic equation $f(\\rho)$ ([equation 4](#eq4))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The largest root $\\lambda$ of $f(\\rho)$ ([equation 4](#eq4)) is given by:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq8'></a>\n",
    "\n",
    "$$\n",
    "\\lambda = \\frac{-p_{1} + \\sqrt{\\Delta}}{2} \\: , \\tag{8}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq9'></a>\n",
    "\n",
    "$$\n",
    "\\Delta = p_{1}^{2} - 4 \\, p_{0} \\: . \\tag{9}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cells below use the equations [8](#eq8) and [9](#eq9) to compute the root $\\lambda$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "delta = p1**2 - 4.*p2*p0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lamb = (-p1 + np.sqrt(delta))/(2.*p2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lambda = 1543.90166\n"
     ]
    }
   ],
   "source": [
    "print 'lambda = %.5f' % lamb"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By substituing $\\lambda$ in [equation 4](#eq4), we can verify that it is a root of $f(\\rho)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f_lamb = p2*(lamb**2) + p1*lamb + p0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(lambda) = -0.00000\n"
     ]
    }
   ],
   "source": [
    "print 'f(lambda) = %.5f' % f_lamb"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='webster-dynamics'></a>\n",
    "\n",
    "* Webster, A. G. 1904. The Dynamics of Particles and of Rigid, Elastic and Fluid Bodies. Universidade de Michigan."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}
